Computing singular locus
I have a bunch of homogeneous polynomials in 5 variables with specific arbitrary (symbolic) non-zero coefficients, i.e. some of them are zero and some of them are non-zero but I don't know the value. For instance:
$$F=a_0x_0f_3(x_1, x_2, x_3)+x_1^2f_2(x_0, \ldots, x_4)+x_0x_1g_3(x_0, x_1,x_2,x_3,x_4)$$
where $f_i, g_j$ are the polynomials with arbitrary non-zero coefficients of degree $i$, $j$, respectively, e.g. say that f_3 has all possible monomials of degree 3 in variables $x_1, x_2, x_3$ with arbitrary coefficients $a_1, a_2, \ldots$
I don't care much which field the coefficients belong to (but if you must know, let it be $\mathbb C$). I want to find the singular locus (i.e. the points $p\in \mathbb P^4$ where all partial derivatives $\frac{\partial F}{\partial x_i}(p)$ of $F$ simultaneously vanish) of one such $F$ in terms of symbolic coefficients and variables $x_i$. Is this something Sagemath can do? If so, can you give me a MWE? For the avoidance of doubt, I am happy to rewrite $F$ above to make the coefficients explicit (e.g. $F=a_0x_0x_1^3+a_1x_0x_1x_2^2+\cdots$).
Thank you.
By "singular", do you mean
points where all derivatves of $F$ wrt to $(x_0,x_1\dots,x_4)$ simultaneously vanish to 0 ?
or possibly points where two or more, but not necessarily all five, of those derivatives vanish ?
or something else ?
In https://www.singular.uni-kl.de/ftp/pu... there are some examples of investigating the singular locus. Singular is a part of Sage. In https://faculty.math.illinois.edu/Mac... one can find some examples of finding the singular locus with Macaulay 2 which can be used in Cocalc and Sage CellServer. Singular locus is also mentioned in https://doc.sagemath.org/html/en/refe... and https://trac.sagemath.org/ticket/3253
@Emmanuel Charpentier Thanks for your question. I mean the first option. I have edited accordingly. My apologies..
@achrzesz Thanks for your message. Is there a way to define in
https://doc.sagemath.org/html/en/refe...z%5E2%20%2D%204x*z%5E3%20%2D
the ideal with arbitrary coefficients instead of specific ones? The problem I find is that I don't want the coefficients to be treated as variables but as 'arbitrary non-zero constants'.
Do you mean
Note however:
(For a better solution see more comments)
@achrzesz Yeah, I think that is what I want, assuming that "J" displays as an ideal. I don't have Sage in this computer but I will check later. PS: I don't get what the warning is about. Is it simply warning you it is slow? I have time :-)
You can always check codes on Sagecell Server (google sagecell). The slownes can be real obstacle. I think that in the books mentioned above you can find faster methods (but I'm not specialist in this field). Toy implementations are good for education but not for investigations. I suppose that the warning appears because you want symbolic coefficients.